Finite Group Databasebeta

← #13159448

Subgroups — #13159448

Order 1728.

1080 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89014C211yesmaximal, characteristic
2C3×#64106C211yesmaximal, characteristic, complemented
3C2×#88800C211yesmaximal, characteristic, complemented
4C2×#88792C211yesmaximal, characteristic, complemented
5#11722180C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592482—31nomaximal
9#10590614—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6290—21no—
15C2×D3×C3⋊Dic3—21no—
16#89014—22no—
17#89014—22no—
18C33×C4⋊C4C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×#3752—21no—
21C3×C4⋊(C3⋊Dic3)—21no—
22C2×C32×C3⋊C8C2211yescharacteristic
23#88800—22no—
24#88792—22no—
25C2×#3830—31no—
26C2×#3795—34no—
27C2×#3754—31no—
28C2×#3746—31no—
29#64106D311yescharacteristic, complemented
30C3×#897—34no—
31C2×#3719—34no—
32C2×#3715—34no—
33C32×#102—31no—
34#63647—31no—
35#63637—31no—
36#63637—34no—
37#63637—64no—
38C3×#809D314yescomplemented
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3×C3⋊D3D411yescharacteristic
41C22×C32×D3—21no—
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45#6308—21no—
46#6308D412yes—
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×C4×C3⋊D3—21no—
50C3×C4×C3⋊D3—21no—
51C3×#634—21no—
52C3×#634—21no—
53C32×D12—21no—
54C32×D12D412yes—
55#6290—24no—
56D3×C3⋊Dic3—24no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C32×C3⋊C8—21no—
59#5047—91no—
60#4783—94no—
61C2×C32×D4—31no—
62C2×#638—61no—
63C22×C3⋊Dic3—61no—
64#3830—62no—
65#3830—62no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×#634—31no—
68C2×#634—34no—
69C2×#634—64no—
70C2×#634C2×D311yescharacteristic
71C2×C3×D12C2×D314yes—
72C2×C3×C4×D3—34no—
73C2×C3×Dic6—34no—
74C2×#625—64no—
75C2×D3×Dic3—64no—
76#3795—68no—
77#3795—68no—
78C2×C32×C8—31no—
79C32×C4⋊C4—24no—
80C32×C4⋊C4C2×D314yes—
81C32×C4⋊C4C2×D311yescharacteristic
82#3754—62no—
83#3752—21no—
84C4⋊(C3⋊Dic3)—21no—
85C2×C3⋊(C3⋊C8)—31no—
86C2×C3⋊(C3⋊C8)—34no—
87C2×C3⋊(C3⋊C8)—64no—
88#3746—62no—
89C3×(C2×D3)⋊C4—64no—
90C3×C4⋊Dic3—64no—
91C2×C3×C3⋊C8C2×D314yes—
92#3719—68no—
93#3715—68no—
94C22×C33C2×D411yescharacteristic
95C2×C3×C3⋊D3—21no—
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3—21no—
98C2×C32×D3—22no—
99C2×C32×D3—41no—
100C2×C32×D3—41no—
101C33×C4—41no—
102C33×C4C2×D411yescharacteristic
103C33×C4C2×D411yescharacteristic
104C3⋊(C3⋊Dic3)—22no—
105C3×C3⋊Dic3—21no—
106C3×C3⋊Dic3—41no—
107C2×#237—94no—
108C2×#235—91no—
109C3×C22⋊Q8—91no—
110C2×#216—91no—
111C2×#215—91no—
112C2×#214—94no—
113C2×Dic12—94no—
114#897—34no—
115C3×#102—34no—
116#830—91no—
117(C2×D3)⋊C8—94no—
118#810—94no—
119#810—91no—
120#810—184no—
121#809C3⋊D311yescharacteristic, complemented
122C23×C32—61no—
123C22×C3⋊D3#6711yescharacteristic
124C22×C3×D3—24no—
125C22×C3×D3—34no—
126C32×D4—32no—
127C32×D4—61no—
128C2×C32×C4—21no—
129C2×C32×C4—24no—
130C2×C32×C4—24no—
131C2×C32×C4—44no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D311yescharacteristic
134#638—64no—
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—64no—
137C2×C3⋊Dic3—61no—
138C2×C3⋊Dic3—64no—
139C2×C3⋊Dic3—124no—
140C2×C3⋊Dic3#6711yescharacteristic
141C4×C3⋊D3—21no—
142C4×C3⋊D3—21no—
143#634—21no—
144#634—21no—
145#634—32no—
146#634—38no—
147#634—68no—
148#634—61no—
149#634—64no—
150#634—124no—
151C2×C3×Dic3—34no—
152C2×C3×Dic3—64no—
153C3×D12—24no—
154C3×D12D1218yes—
155C3×C4×D3—64no—
156C3×C4×D3—64no—
157C3×Dic6—64no—
158C3×Dic6—64no—
159#625—616no—
160D3×Dic3—616no—
161C32×C8—61no—
162C3⋊(C3⋊C8)—61no—
163C3⋊(C3⋊C8)—64no—
164C3⋊(C3⋊C8)—124no—
165C3×C3⋊C8—24no—
166#452—271noSylow
167C2×C33#12011yescharacteristic
168C2×C33#13511yescharacteristic
169C2×C33#13711yescharacteristic
170C3×C3⋊D3—41no—
171C32×D3—42no—
172C2×C3×Q8—91no—
173C2×C3×D4—34no—
174C22×C3×C4—91no—
175C2×#67—181no—
176C2×#67—184no—
177C22×Dic3—184no—
178#237—188no—
179#237—188no—
180#235—182no—
181#235—182no—
182C2×D12C2×C3⋊D311yescharacteristic
183C2×C4×D3—34no—
184C2×C4×D3—181no—
185C2×Dic6—34no—
186C2×Dic6—94no—
187C2×Dic6—91no—
188C2×Dic6—184no—
189C2×C3×C8—34no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—181no—
192C3×C4⋊C4D3214yes—
193C3×C4⋊C4C2×C3⋊D311yescharacteristic
194C3×C22⋊C4—181no—
195#216—182no—
196#215—182no—
197#214—188no—
198(C2×D3)⋊C4—64no—
199C4⋊Dic3—64no—
200C2×C3⋊C8—94no—
201C2×C3⋊C8—91no—
202C2×C3⋊C8—184no—
203C2×C3⋊C8C2×C3⋊D311yescharacteristic
204Dic12—188no—
205C22×C32—24no—
206C22×C32—62no—
207C22×C32—121no—
208C22×C32—121no—
209C22×C32D3×D414yes—
210C22×C32C2×#6711yescharacteristic
211C2×C3⋊D3—21no—
212C2×C3⋊D3—21no—
213C2×C3⋊D3—21no—
214C2×C3×D3—28no—
215C2×C3×D3—44no—
216C2×C3×D3—44no—
217C2×C3×D3—64no—
218C2×C3×D3—64no—
219C2×C3×D3—64no—
220C32×C4—24no—
221C32×C4—24no—
222C32×C4—41no—
223C32×C4—44no—
224C32×C4—84no—
225C32×C4C2×D1214yes—
226C32×C4D3×D414yes—
227C32×C4D3×D411yescharacteristic
228C32×C4D3×D411yescharacteristic
229C3⋊Dic3—21no—
230C3⋊Dic3—41no—
231C3⋊Dic3—62no—
232C3⋊Dic3—68no—
233C3⋊Dic3—128no—
234C3×Dic3—64no—
235C3×Dic3—124no—
236C2×#41—271no—
237C2×Q16—271no—
238C2×SD16—271no—
239C22⋊Q8—91no—
240#103—271no—
241#102—31no—
242C22⋊C8—271no—
243C33#45211yescharacteristic, complemented, Sylow
244C23×C3—64no—
245C23×C3—91no—
246C22×D3—21no—
247C22×D3—34no—
248C3×Q8—181no—
249C3×Q8—181no—
250C3×D4—38no—
251C3×D4—64no—
252C2×C3×C4—24no—
253C2×C3×C4—21no—
254C2×C3×C4—24no—
255C2×C3×C4—44no—
256C2×C3×C4—91no—
257C2×C3×C4—181no—
258C2×C3×C4—181no—
259C2×C3×C4—181no—
260C2×C3×C4C2×D3214yes—
261C2×C3×C4C22×C3⋊D311yescharacteristic
262#67—184no—
263#67—1816no—
264C2×Dic3—34no—
265C2×Dic3—64no—
266C2×Dic3—1816no—
267C2×Dic3—184no—
268C2×Dic3—181no—
269C2×Dic3—364no—
270D12—21no—
271D12C4⋊(C3⋊D3)12yes—
272C4×D3—64no—
273C4×D3—64no—
274C4×D3—184no—
275Dic6—64no—
276Dic6—64no—
277Dic6—98no—
278Dic6—92no—
279Dic6—188no—
280Dic6—184no—
281Dic6—181no—
282Dic6—364no—
283C3×C8—64no—
284C3⋊C8—21no—
285C3⋊C8—184no—
286C3⋊C8—181no—
287C3⋊C8—364no—
288C2×C32—24no—
289C2×C32—24no—
290C2×C32—24no—
291C2×C32—122no—
292C2×C32#88214yes—
293C2×C32#91214yes—
294C2×C32#91514yes—
295C2×C32#93711yescharacteristic
296C2×C32#94211yescharacteristic
297C2×C32#95011yescharacteristic
298C3⋊D3—41no—
299C3×D3—48no—
300C3×D3—124no—
301#41—542no—
302#41—542no—
303C2×Q8—91no—
304C2×Q8—271no—
305C2×D4—31no—
306C2×D4—541no—
307C22×C4—91no—
308C22×C4—541no—
309Q16—542no—
310SD16—542no—
311C2×C8—31no—
312C2×C8—271no—
313C4⋊C4—181no—
314C4⋊C4D3×C3⋊D311yescharacteristic
315C22⋊C4—181no—
316C22×C3—24no—
317C22×C3—68no—
318C22×C3—124no—
319C22×C3—124no—
320C22×C3—181no—
321C22×C3—181no—
322C22×C3—181no—
323C22×C3D3×#6714yes—
324C22×C3D4×C3⋊D311yescharacteristic
325C2×D3—22no—
326C2×D3—41no—
327C2×D3—41no—
328C2×D3—64no—
329C2×D3—64no—
330C2×D3—64no—
331C3×C4—24no—
332C3×C4—24no—
333C3×C4—41no—
334C3×C4—44no—
335C3×C4—84no—
336C3×C4—181no—
337C3×C4—361no—
338C3×C4D3×D1214yes—
339C3×C4#380214yes—
340C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
341C3×C4D4×C3⋊D311yescharacteristic
342Dic3—64no—
343Dic3—124no—
344Dic3—182no—
345Dic3—188no—
346Dic3—368no—
347C32—24no—
348C32#478314yescomplemented
349C32#504711yescharacteristic, complemented
350C23—61no—
351C23—91no—
352Q8—181no—
353Q8—181no—
354Q8—272no—
355Q8—541no—
356D4—32no—
357D4—61no—
358D4—544no—
359C2×C4—21no—
360C2×C4—91no—
361C2×C4—181no—
362C2×C4—181no—
363C2×C4—181no—
364C2×C4—544no—
365C2×C4—541no—
366C2×C4C2×D3×C3⋊D311yescharacteristic
367C8—61no—
368C8—541no—
369C2×C3—24no—
370C2×C3—24no—
371C2×C3—24no—
372C2×C3—128no—
373C2×C3—361no—
374C2×C3#6392214yes—
375C2×C3#6393314yes—
376C2×C3#6394614yes—
377C2×C3#6409111yescharacteristic
378C2×C3#6412111yescharacteristic
379C2×C3#6412411yescharacteristic
380D3—42no—
381D3—124no—
382C22—62no—
383C22—121no—
384C22—121no—
385C22—181no—
386C22—181no—
387C22—181no—
388C22C3⋊D3×#6711yescharacteristic
389C4—41no—
390C4—181no—
391C4—361no—
392C4—542no—
393C4D3×C4⋊(C3⋊D3)11yescharacteristic
394C4#8902611yescharacteristic
395C3—24no—
396C3#1059061414yescomplemented
397C3#1059248211yescharacteristic, complemented
398C2—122no—
399C2—361no—
400C2#1172354111yescharacteristic
401C2#1172355911yescharacteristic
402C2#1172358411yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×#89014C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#452
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×#64106C2semidirect
3. C2×#88800C2semidirect
4. C2×#88792C2semidirect
5. #11722180C2semidirect
6. C32×#809C2semidirect
7. #11722023C2semidirect
29. #64106D3semidirect
38. C3×#809D3semidirect
121. #809C3⋊D3semidirect
243. C33#452semidirect
348. C32#4783semidirect
349. C32#5047semidirect
396. C3#10590614semidirect
397. C3#10592482semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#890142
C3×#641062
C2×#888002
C2×#887922
#117221802
C32×#8092
#117220232
#105924823
#105906143